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TC4 welded by the electron beam process
Materials Science and Engineering A 425 (2006) 255–259
Interfacial microstructure and strength of the dissimilar joint Ti3Al/TC4 welded by the electron beam process Hongtao Zhang ∗ , Peng He, Jicai Feng, Huiqiang Wu State Key Laboratory of Advanced Welding Production Technology, Harbin Institute of Technology, Harbin 150001, Heilongjiang Province, PR China Received 9 December 2005; accepted 21 March 2006
Abstract Dissimilar weld joints in Ti3 Al/TC4 were welded using the electron beam (EB) process. The microstructure evolution characterizations of the joints was investigated by means of OM, SEM, XRD, TEM and the tensile strengths of the joints were tested. The microstructure of the weld metal of every joint was identical. The structures were characterized by martensite, appearing coarse equiaxed grains. There existed grain coarsening and martensitic transformation in Ti3 Al/HAZ and TC4/HAZ. With the increase of heat input, the grain size was significantly raised, yet the composition of the weld metal was independent of heat input. The highest tensile strength of the joints can reach 831 MPa, equaled almost to 92% of that of Ti3 Al-based alloy. © 2006 Elsevier B.V. All rights reserved. Keywords: Electron beam welding; Titanium alloy; Titanium aluminides
1. Introduction Electron beam welding (EBW) is a fusion joining process that dose not require the addition of filler metal. Moreover, welding is carried out in vacuum to avoid oxidation. EBW is also an excellent process for joining refractory metals and their dissimilar combinations because of the high energy density [1,2]. Titanium alloy TC4 (Ti–6Al–4V) and Ti3 Al-based alloy are widely used in the aerospace and military industries for their remarkable strengths, weight ratio, and resistance to high temperature creep. At temperatures above 350 ◦ C, and particularly in a molten state, titanium alloys are known to be very reactive towards most atmospheric gases such as oxygen, nitrogen or hydrogen [3,4]. Thomas and Ramachandra [5] indicated that EBW is the most popular method to join Ti–6Al–4V with high quality for its high density of energy and vacuum environment. As to the weldability of Ti3 Al-based alloy, some work has been done to concluded that the resistance to solidification cracking was good when welded by electron beam [6–10]. To date, there are rare reports on the joint between Ti3 Al-based alloy and titanium alloy.
∗
Corresponding author. Tel.: +86 451 86412974; fax: +86 451 86418146. E-mail address: [email protected] (H. Zhang).
0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.03.048
The present study is to discuss the weldability of dissimilar joining between Ti3 Al and TC4. The emphasis has been placed on the analysis of the evolution of the joint microstructure. The fracture mechanism and the effect of heat input on microstructure and tensile strength are also discussed.
2. Experimental The two alloys used for electron beam welding are all hot-rolled sheet materials, with a thickness of 3.1 mm. The microstructure of the Ti3 Al-based alloy is composed of ␣2 +  + o three-phase equiaxial grains (shown in Fig. 1) and the microstructure of the as-received Ti–6Al–4V consists of small percentage of  phase distributed at the elongated ␣ grain boundaries (shown in Fig. 2). The chemical composition by weight percentage of two base metals are listed in Table 1. Before welding, any oxide layer was removed from the surfaces and the specimens were cleaned with a specific acid solution (35 ml HNO3 , 5 ml HF, and 60 ml glycerol). All welding experiments were conducted using TECHMETA MEDARD 45 model pulsed electron beam welding machine with accelerating voltage 55 kV, and the electron currents varying from 9 to 18 mA, and the welding speed was selected as 400 mm/min. The welded samples were cut into small metallographic specimens and then conventional titanium metallographic proce-
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H. Zhang et al. / Materials Science and Engineering A 425 (2006) 255–259 Table 1 The chemical composition by weight percentage of Ti3 Al-based alloy and TC4 alloy Ti3 Al
Fig. 1. Microstructure of Ti3 Al base metal.
TC4
Elements
wt.%
Elements
wt.%
Ti Al Nb Fe Mo Cr Si Cu Mn
Balance 9.83 26.10 0.07 0.07 0.09 0.07 0.02 0.02
Ti Al V Fe C N H O –
Balance 5.82 3.99 ≤0.05 0.019 0.0032 0.0007 0.063 –
zone, and then thinned by plasma-pump to observe with transmission electron microscopy (TEM). X-ray diffraction was used for primary phase identification of the fusion zone. Microhardness values were obtained using a microhardness tester with a load of 100 g, and a load time of 15 s. Tensile tests were conducted on the welded specimens to find their strengths. The fracture modes of the specimens were investigated by scanning electron microscopy (SEM).
Fig. 2. Microstructure of TC4 base metal.
dures were used to prepare the metallographic specimens. After the metallographic specimens were etched by a Kroll solution (100 ml H2 O, 3 ml HF, and 5 ml HNO3 ), microstructure characterization was performed using optical microscopy (OM), scanning electron microscopy (SEM). A slice was cut from fusion
Fig. 3. The profile of the joint by EBW.
Fig. 4. OM microstructure of fusion zone (a) upper weld; (b) lower weld.
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Fig. 5. HAZ microstructure of Ti3 Al electron beam welded joint: (a) near-HAZ; (b) far-HAZ.
3. Results and discussions 3.1. The microstructure characterization of the joint Well-formed surfaces of the welded samples were gained by EBW independence of processing parameters, while the surfaces of the welds were slightly concave. A asymmetric funnel-like joint can be seen from Fig. 3 due to the different thermophysical parameters of two different base metals. The fusion zone of the EB weld is formed quite narrow in 0.8–2.2 mm width, and is clearly distinguished from the two base metals. Different grain textures can be found in Fig. 4 between the upper weld and the lower weld owing to the diversity of heat transfer direction. When welding started but before penetration, the heat transferred along cross-direction and depth direction, consequently generated the columnar grains whose orientation was perpendicular to the boundary between the fusion zone and the heat affected zone (HAZ). After a narrow, deeply penetrating vapour cavity was formed by the electron beam, the input heat mainly transferred along the cross-direction, and thus generated coarse equiaxed grains. No obvious second phase was observed in the weld and just the solidification crystals were apparent. According to Ref. [11], the HAZ of the Ti3 Al-based alloy can be divided into two regions, a “far-HAZ” and a “near-HAZ”, as shown in Fig. 5. This microstructure feature is mainly caused by the different weld thermal cycles at two different zone. Optical microscopy micrograph shows that the HAZ in the vicinity of molten boundary of titanium alloy TC4 consists of mainly ␣ martensite (Fig. 6). The acicular ␣ in the HAZ was attributed to rapid cooling of the weld metal. The grain size of the  matrix has been increased greatly by the thermal cycle of the welding. This result consists with titanium welded by laser beam [12].
Fig. 6. HAZ microstructure of TC4 alloy.
the weld metal predominantly contains B2 phase when joining Ti3 Al-based alloy by electron beam, separately. But in this experiment, the microstructure of the weld metal mainly consists of ␣ martensite. This result may be on account of the dilution of the elements which can suppress /␣ phase transformation when cooling from high temperature. This dilution mainly attribute to the addition of TC4 alloy in the fusion metal which contains less  phase stabilizer than that of Ti3 Al-based alloy.
3.2. XRD and TEM analysis of the fusion zone XRD was used to determine the different phases that developed in the fusion zone. In the diffraction spectra of weld metal, only the diffractive peaks of ␣ phase could be found (Fig. 7). Fig. 8 shows that the weld metal is mainly composed of acicular ␣ martensite with random orientations and a little residual  phase. According to Ref. [6], the cooling rate appears to be fast enough to suppress  to ␣2 phase transformation and
Fig. 7. XRD spectra of weld metal.
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H. Zhang et al. / Materials Science and Engineering A 425 (2006) 255–259
Fig. 10. Elements distribution of EBW joint.
Fig. 8. TEM microstructure of weld metal.
3.3. Hardness measurement Fig. 9 shows the microhardness distribution of the dissimilar metal joint. The hardness of the HAZ is higher than that of the BM. These results were consistent with the microstructure observations, that is, the acicular martensite product and the hard phase ␣2 rapidly increase in the HAZ of TC4 alloy and Ti3 Al-based alloy, respectively. The microhardness in the fusion zone drops acutely as the distance from the fusion boundary near titanium alloy TC4 increases. This trend of microhardness distribution may result in the different amount of martensite at different position in fusion zone which might arise from the different content of beta phase stabilizer in transverse section (Fig. 10), such as Nb. The beta stabilizer does not only enrich the beta phase but also reduce the amount of martensite. The different content of the alloying element V in the weld metal also may have a function on the microhardness distribution for its solution strengthen in titanium alloys.
Fig. 9. Microhardness distribution of EBW joint.
Fig. 11. Tensile strengths and welding parameters.
3.4. Tensile properties Through changing the EBW currents, specimens of different heat inputs could be obtained. No composition change in weld metal was found after qualitative analysis. Transverse tensile tests were conducted on these welded specimens to find their strengths at room temperature. Fig. 11 shows a trend of decreasing tensile strengths with increasing theoretical heat inputs. This
Fig. 12. The tensile fracture face of the electron beam welded joint.
H. Zhang et al. / Materials Science and Engineering A 425 (2006) 255–259
259
Fig. 13. Optical microstructure of the weld metal in different heat inputs: (a) upper weld; (b) lower weld (E = 90.75 J/mm); (c) upper weld; (d) lower weld (E = 148.5 J/mm).
indicated that there was a negative relationship between the heat input and the weldment strength. At the same time, the highest tensile strength of the joint can reach 831 MPa, equaled almost to 92% of that of Ti3 Al-based alloy. It is important to mention that the fracture was located within the fusion zone regardless of the welding parameters. The fracture of the joint was brittle-like fracture (Fig. 12), implying that ductility of the alloy was deteriorated at the fusion zone upon welding. It is well established that the microstructure of the joints affect the tensile strength critically. The influence of the heat inputs on the microstructure can be clearly realized on the grain size variation with different heat inputs in Fig. 13. The higher the heat inputs, the more dwelling time at liquid temperature could accelerate the growing of the grain size and deteriorate the tensile strength of the weldment.
hardness increasing in HAZ of two base metals and a varying hardness in fusion zone because of the different quantity of martensite in different position which arised by the nonhomogeneous distribution of the beta phase stabilizer. (3) Grain size affects the tensile strength of the dissimilar joint critically and this factor is mainly related to the heat inputs during welding processing. A good joint with high tensile strength 831 MPa can be presented by selecting small heat input which can penetrate the butted joint.
4. Conclusions
References
Based on the experimental results and discussions, conclusions were drawn as follows: (1) The dissimilar joint between Ti3 Al and TC4 alloys was welded by EBW by single pass without filler material. The profile of transverse section in weldments is asymmetric funnel-like. Because of the difference of the heat transfer, columnar grains were formed in the upper fusion zone and did the equiaxed grains in the lower fusion zone. (2) Microstructure analysis of the joint has shown the weld metal was mainly composed of martensite alpha prime. In two different metal HAZ, grain growing has been detected. The microhardness of the joint distribution also has shown
Acknowledgement The research is sponsored by the National Natural Science Foundation of China (No. 50505008) and Program for New Century Excellent Talents in University.
[1] M. Ahmad, J.I. Akhter, M. Akhtar, J. Alloys Compd. 390 (1–2) (2005) 88–93. [2] Z. Sun, R. Karppi, J. Mater. Process. Technol. 59 (3) (1996) 257–267. [3] J.L. Barreda, F. Santamara, X. Azpiroz, Vaccum 62 (2–3) (2001) 143–150. [4] J.L. Barreda, F. Santamaria, Vacuum 62 (15) (2001) 143–150. [5] G. Thomas, V. Ramachandra, Weld. J. 68 (8) (1989) 336–341. [6] P.L. Treadgill, Mater. Sci. Eng. 192/193 (A) (1995) 640–646. [7] A.P. Wu, Intermetallics (10) (2002) 647–652. [8] W.A. Baeslack III, T. Mascorella, T.J. Kelly, Weldability of a titanium aluminide, Weld. J. 68 (12) (1989) 483–498. [9] S.A. David, J.A. Horton, G.M. Goodwin, D.H. Phillips, R.W. Reed, Weld. J. 69 (4) (1990) 133–140. [10] G.S. Martin, C.E. Albright, T.A. Jones, Weld. J. 74 (2) (1995) 77–82. [11] J. Feng, H. Wu, J. He, B. Zhang, Mater. Charact. 54 (2) (2005) 99–105. [12] L.W. Tsay, C.Y. Tsay, Int. J. Fatigue 19 (10) (1997) 713–720.
Interfacial microstructure and strength of the dissimilar joint Ti3Al/TC4 welded by the electron beam process Hongtao Zhang ∗ , Peng He, Jicai Feng, Huiqiang Wu State Key Laboratory of Advanced Welding Production Technology, Harbin Institute of Technology, Harbin 150001, Heilongjiang Province, PR China Received 9 December 2005; accepted 21 March 2006
Abstract Dissimilar weld joints in Ti3 Al/TC4 were welded using the electron beam (EB) process. The microstructure evolution characterizations of the joints was investigated by means of OM, SEM, XRD, TEM and the tensile strengths of the joints were tested. The microstructure of the weld metal of every joint was identical. The structures were characterized by martensite, appearing coarse equiaxed grains. There existed grain coarsening and martensitic transformation in Ti3 Al/HAZ and TC4/HAZ. With the increase of heat input, the grain size was significantly raised, yet the composition of the weld metal was independent of heat input. The highest tensile strength of the joints can reach 831 MPa, equaled almost to 92% of that of Ti3 Al-based alloy. © 2006 Elsevier B.V. All rights reserved. Keywords: Electron beam welding; Titanium alloy; Titanium aluminides
1. Introduction Electron beam welding (EBW) is a fusion joining process that dose not require the addition of filler metal. Moreover, welding is carried out in vacuum to avoid oxidation. EBW is also an excellent process for joining refractory metals and their dissimilar combinations because of the high energy density [1,2]. Titanium alloy TC4 (Ti–6Al–4V) and Ti3 Al-based alloy are widely used in the aerospace and military industries for their remarkable strengths, weight ratio, and resistance to high temperature creep. At temperatures above 350 ◦ C, and particularly in a molten state, titanium alloys are known to be very reactive towards most atmospheric gases such as oxygen, nitrogen or hydrogen [3,4]. Thomas and Ramachandra [5] indicated that EBW is the most popular method to join Ti–6Al–4V with high quality for its high density of energy and vacuum environment. As to the weldability of Ti3 Al-based alloy, some work has been done to concluded that the resistance to solidification cracking was good when welded by electron beam [6–10]. To date, there are rare reports on the joint between Ti3 Al-based alloy and titanium alloy.
∗
Corresponding author. Tel.: +86 451 86412974; fax: +86 451 86418146. E-mail address: [email protected] (H. Zhang).
0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.03.048
The present study is to discuss the weldability of dissimilar joining between Ti3 Al and TC4. The emphasis has been placed on the analysis of the evolution of the joint microstructure. The fracture mechanism and the effect of heat input on microstructure and tensile strength are also discussed.
2. Experimental The two alloys used for electron beam welding are all hot-rolled sheet materials, with a thickness of 3.1 mm. The microstructure of the Ti3 Al-based alloy is composed of ␣2 +  + o three-phase equiaxial grains (shown in Fig. 1) and the microstructure of the as-received Ti–6Al–4V consists of small percentage of  phase distributed at the elongated ␣ grain boundaries (shown in Fig. 2). The chemical composition by weight percentage of two base metals are listed in Table 1. Before welding, any oxide layer was removed from the surfaces and the specimens were cleaned with a specific acid solution (35 ml HNO3 , 5 ml HF, and 60 ml glycerol). All welding experiments were conducted using TECHMETA MEDARD 45 model pulsed electron beam welding machine with accelerating voltage 55 kV, and the electron currents varying from 9 to 18 mA, and the welding speed was selected as 400 mm/min. The welded samples were cut into small metallographic specimens and then conventional titanium metallographic proce-
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H. Zhang et al. / Materials Science and Engineering A 425 (2006) 255–259 Table 1 The chemical composition by weight percentage of Ti3 Al-based alloy and TC4 alloy Ti3 Al
Fig. 1. Microstructure of Ti3 Al base metal.
TC4
Elements
wt.%
Elements
wt.%
Ti Al Nb Fe Mo Cr Si Cu Mn
Balance 9.83 26.10 0.07 0.07 0.09 0.07 0.02 0.02
Ti Al V Fe C N H O –
Balance 5.82 3.99 ≤0.05 0.019 0.0032 0.0007 0.063 –
zone, and then thinned by plasma-pump to observe with transmission electron microscopy (TEM). X-ray diffraction was used for primary phase identification of the fusion zone. Microhardness values were obtained using a microhardness tester with a load of 100 g, and a load time of 15 s. Tensile tests were conducted on the welded specimens to find their strengths. The fracture modes of the specimens were investigated by scanning electron microscopy (SEM).
Fig. 2. Microstructure of TC4 base metal.
dures were used to prepare the metallographic specimens. After the metallographic specimens were etched by a Kroll solution (100 ml H2 O, 3 ml HF, and 5 ml HNO3 ), microstructure characterization was performed using optical microscopy (OM), scanning electron microscopy (SEM). A slice was cut from fusion
Fig. 3. The profile of the joint by EBW.
Fig. 4. OM microstructure of fusion zone (a) upper weld; (b) lower weld.
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Fig. 5. HAZ microstructure of Ti3 Al electron beam welded joint: (a) near-HAZ; (b) far-HAZ.
3. Results and discussions 3.1. The microstructure characterization of the joint Well-formed surfaces of the welded samples were gained by EBW independence of processing parameters, while the surfaces of the welds were slightly concave. A asymmetric funnel-like joint can be seen from Fig. 3 due to the different thermophysical parameters of two different base metals. The fusion zone of the EB weld is formed quite narrow in 0.8–2.2 mm width, and is clearly distinguished from the two base metals. Different grain textures can be found in Fig. 4 between the upper weld and the lower weld owing to the diversity of heat transfer direction. When welding started but before penetration, the heat transferred along cross-direction and depth direction, consequently generated the columnar grains whose orientation was perpendicular to the boundary between the fusion zone and the heat affected zone (HAZ). After a narrow, deeply penetrating vapour cavity was formed by the electron beam, the input heat mainly transferred along the cross-direction, and thus generated coarse equiaxed grains. No obvious second phase was observed in the weld and just the solidification crystals were apparent. According to Ref. [11], the HAZ of the Ti3 Al-based alloy can be divided into two regions, a “far-HAZ” and a “near-HAZ”, as shown in Fig. 5. This microstructure feature is mainly caused by the different weld thermal cycles at two different zone. Optical microscopy micrograph shows that the HAZ in the vicinity of molten boundary of titanium alloy TC4 consists of mainly ␣ martensite (Fig. 6). The acicular ␣ in the HAZ was attributed to rapid cooling of the weld metal. The grain size of the  matrix has been increased greatly by the thermal cycle of the welding. This result consists with titanium welded by laser beam [12].
Fig. 6. HAZ microstructure of TC4 alloy.
the weld metal predominantly contains B2 phase when joining Ti3 Al-based alloy by electron beam, separately. But in this experiment, the microstructure of the weld metal mainly consists of ␣ martensite. This result may be on account of the dilution of the elements which can suppress /␣ phase transformation when cooling from high temperature. This dilution mainly attribute to the addition of TC4 alloy in the fusion metal which contains less  phase stabilizer than that of Ti3 Al-based alloy.
3.2. XRD and TEM analysis of the fusion zone XRD was used to determine the different phases that developed in the fusion zone. In the diffraction spectra of weld metal, only the diffractive peaks of ␣ phase could be found (Fig. 7). Fig. 8 shows that the weld metal is mainly composed of acicular ␣ martensite with random orientations and a little residual  phase. According to Ref. [6], the cooling rate appears to be fast enough to suppress  to ␣2 phase transformation and
Fig. 7. XRD spectra of weld metal.
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H. Zhang et al. / Materials Science and Engineering A 425 (2006) 255–259
Fig. 10. Elements distribution of EBW joint.
Fig. 8. TEM microstructure of weld metal.
3.3. Hardness measurement Fig. 9 shows the microhardness distribution of the dissimilar metal joint. The hardness of the HAZ is higher than that of the BM. These results were consistent with the microstructure observations, that is, the acicular martensite product and the hard phase ␣2 rapidly increase in the HAZ of TC4 alloy and Ti3 Al-based alloy, respectively. The microhardness in the fusion zone drops acutely as the distance from the fusion boundary near titanium alloy TC4 increases. This trend of microhardness distribution may result in the different amount of martensite at different position in fusion zone which might arise from the different content of beta phase stabilizer in transverse section (Fig. 10), such as Nb. The beta stabilizer does not only enrich the beta phase but also reduce the amount of martensite. The different content of the alloying element V in the weld metal also may have a function on the microhardness distribution for its solution strengthen in titanium alloys.
Fig. 9. Microhardness distribution of EBW joint.
Fig. 11. Tensile strengths and welding parameters.
3.4. Tensile properties Through changing the EBW currents, specimens of different heat inputs could be obtained. No composition change in weld metal was found after qualitative analysis. Transverse tensile tests were conducted on these welded specimens to find their strengths at room temperature. Fig. 11 shows a trend of decreasing tensile strengths with increasing theoretical heat inputs. This
Fig. 12. The tensile fracture face of the electron beam welded joint.
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259
Fig. 13. Optical microstructure of the weld metal in different heat inputs: (a) upper weld; (b) lower weld (E = 90.75 J/mm); (c) upper weld; (d) lower weld (E = 148.5 J/mm).
indicated that there was a negative relationship between the heat input and the weldment strength. At the same time, the highest tensile strength of the joint can reach 831 MPa, equaled almost to 92% of that of Ti3 Al-based alloy. It is important to mention that the fracture was located within the fusion zone regardless of the welding parameters. The fracture of the joint was brittle-like fracture (Fig. 12), implying that ductility of the alloy was deteriorated at the fusion zone upon welding. It is well established that the microstructure of the joints affect the tensile strength critically. The influence of the heat inputs on the microstructure can be clearly realized on the grain size variation with different heat inputs in Fig. 13. The higher the heat inputs, the more dwelling time at liquid temperature could accelerate the growing of the grain size and deteriorate the tensile strength of the weldment.
hardness increasing in HAZ of two base metals and a varying hardness in fusion zone because of the different quantity of martensite in different position which arised by the nonhomogeneous distribution of the beta phase stabilizer. (3) Grain size affects the tensile strength of the dissimilar joint critically and this factor is mainly related to the heat inputs during welding processing. A good joint with high tensile strength 831 MPa can be presented by selecting small heat input which can penetrate the butted joint.
4. Conclusions
References
Based on the experimental results and discussions, conclusions were drawn as follows: (1) The dissimilar joint between Ti3 Al and TC4 alloys was welded by EBW by single pass without filler material. The profile of transverse section in weldments is asymmetric funnel-like. Because of the difference of the heat transfer, columnar grains were formed in the upper fusion zone and did the equiaxed grains in the lower fusion zone. (2) Microstructure analysis of the joint has shown the weld metal was mainly composed of martensite alpha prime. In two different metal HAZ, grain growing has been detected. The microhardness of the joint distribution also has shown
Acknowledgement The research is sponsored by the National Natural Science Foundation of China (No. 50505008) and Program for New Century Excellent Talents in University.
[1] M. Ahmad, J.I. Akhter, M. Akhtar, J. Alloys Compd. 390 (1–2) (2005) 88–93. [2] Z. Sun, R. Karppi, J. Mater. Process. Technol. 59 (3) (1996) 257–267. [3] J.L. Barreda, F. Santamara, X. Azpiroz, Vaccum 62 (2–3) (2001) 143–150. [4] J.L. Barreda, F. Santamaria, Vacuum 62 (15) (2001) 143–150. [5] G. Thomas, V. Ramachandra, Weld. J. 68 (8) (1989) 336–341. [6] P.L. Treadgill, Mater. Sci. Eng. 192/193 (A) (1995) 640–646. [7] A.P. Wu, Intermetallics (10) (2002) 647–652. [8] W.A. Baeslack III, T. Mascorella, T.J. Kelly, Weldability of a titanium aluminide, Weld. J. 68 (12) (1989) 483–498. [9] S.A. David, J.A. Horton, G.M. Goodwin, D.H. Phillips, R.W. Reed, Weld. J. 69 (4) (1990) 133–140. [10] G.S. Martin, C.E. Albright, T.A. Jones, Weld. J. 74 (2) (1995) 77–82. [11] J. Feng, H. Wu, J. He, B. Zhang, Mater. Charact. 54 (2) (2005) 99–105. [12] L.W. Tsay, C.Y. Tsay, Int. J. Fatigue 19 (10) (1997) 713–720.